1,203 research outputs found
Approximating the Spectrum of a Graph
The spectrum of a network or graph with adjacency matrix ,
consists of the eigenvalues of the normalized Laplacian . This set of eigenvalues encapsulates many aspects of the structure
of the graph, including the extent to which the graph posses community
structures at multiple scales. We study the problem of approximating the
spectrum , of in the regime where the graph is too
large to explicitly calculate the spectrum. We present a sublinear time
algorithm that, given the ability to query a random node in the graph and
select a random neighbor of a given node, computes a succinct representation of
an approximation , such that . Our algorithm has query complexity and running time ,
independent of the size of the graph, . We demonstrate the practical
viability of our algorithm on 15 different real-world graphs from the Stanford
Large Network Dataset Collection, including social networks, academic
collaboration graphs, and road networks. For the smallest of these graphs, we
are able to validate the accuracy of our algorithm by explicitly calculating
the true spectrum; for the larger graphs, such a calculation is computationally
prohibitive.
In addition we study the implications of our algorithm to property testing in
the bounded degree graph model
Vertex covering with monochromatic pieces of few colours
In 1995, Erd\H{o}s and Gy\'arf\'as proved that in every -colouring of the
edges of , there is a vertex cover by monochromatic paths of
the same colour, which is optimal up to a constant factor. The main goal of
this paper is to study the natural multi-colour generalization of this problem:
given two positive integers , what is the smallest number
such that in every colouring of the edges of with
colours, there exists a vertex cover of by
monochromatic paths using altogether at most different colours? For fixed
integers and as , we prove that , where is the chromatic number of
the Kneser gr aph . More generally, if one replaces by
an arbitrary -vertex graph with fixed independence number , then we
have , where this time around is the
chromatic number of the Kneser hypergraph . This
result is tight in the sense that there exist graphs with independence number
for which . This is in sharp
contrast to the case , where it follows from a result of S\'ark\"ozy
(2012) that depends only on and , but not on
the number of vertices. We obtain similar results for the situation where
instead of using paths, one wants to cover a graph with bounded independence
number by monochromatic cycles, or a complete graph by monochromatic
-regular graphs
Open/Closed String Topology and Moduli Space Actions via Open/Closed Hochschild Actions
In this paper we extend our correlation functions to the open/closed case.
This gives rise to actions of an open/closed version of the Sullivan PROP as
well as an action of the relevant moduli space. There are several unexpected
structures and conditions that arise in this extension which are forced upon us
by considering the open sector. For string topology type operations, one cannot
just consider graphs, but has to take punctures into account and one has to
restrict the underlying Frobenius algebras. In the moduli space, one first has
to pass to a smaller moduli space which is closed under open/closed duality and
then consider covers in order to account for the punctures
Open/closed string topology and moduli space actions via open/closed Hochschild actions
In this paper we extend our correlation functions to the open/closed case.
This gives rise to actions of an open/closed version of the Sullivan PROP as
well as an action of the relevant moduli space. There are several unexpected
structures and conditions that arise in this extension which are forced upon us
by considering the open sector. For string topology type operations, one cannot
just consider graphs, but has to take punctures into account and one has to
restrict the underlying Frobenius algebras. In the moduli space, one first has
to pass to a smaller moduli space which is closed under open/closed duality and
then consider covers in order to account for the punctures
On the quasi-isometric rigidity of graphs of surface groups
Let be a word hyperbolic group with a cyclic JSJ decomposition that
has only rigid vertex groups, which are all fundamental groups of closed
surface groups. We show that any group quasi-isometric to is
abstractly commensurable with .Comment: 54 pages, 10 figures, comments welcom
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